FACTORIZATION PROPERTIES OF OPTIMUM SPACE-TIME PROCESSORS IN NONSTATIONARY ENVIRONMENTS

In this paper, we present a structural analysis of space-time log-likelihood processors (LLP) that applies to arbitrary signal transmission models consisting of N sources, M sensors, a time-varying linear channel, and nonstationary Gaussian source signal and sensor noise processes. Our approach is based on representing the time-varying linear channel as a bounded linear operator L with closed range. By exploiting the properties of such operators and the specific structure of the array covariance function, we show that the classical M-dimensional integral equations defining the LLP can be transformed into equivalent N-dimensional integral equations. As a result, it is always possible to factor the LLP into a cascade of three specialized time-varying subprocessors, namely: a space-time whitening filter, an M-input A'-output unitary beamformer (UB), and an N-input quadratic postprocessor (QPP). This decomposition provides a generalization of conventional results on optimum array processing that were previously derived under the assumption of time-invariant transmission channel and signal statistics. Both the UB and the QPP are given an interpretation and their most important features are indicated. The UB, which is closely related to the generalized inverse of the transmission operator L, is independent of the source signal statistics and maximizes the array gain. Moreover, when N < M, it can be used advantageously to reduce the number of time functions that need to be quadratically processed. The QPP behaves like an LLP for the output of the UB and, therefore, it admits a number of standard realizations, both causal and noncausal. Specializations of the above results to the cases of low and high signal-to-noise ratios are also considered. Finally, to illustrate the theory, several examples of its application to signal models involving time-varying delays are given. © 1990 IEEE